Stochastic optimal control and large deviations in the space of probability measures

I will present problems a stochastic variant of the classic optimal transport problem as well as a large deviation question for a mean field system of interacting particles. We shall see that those problems can be analyzed by means of a Hamilton-Jacobi equation on the space of probability measures. I will then present the main challenge on such equations as well as the current known techniques to address them. In particular, I will show how the notion of relaxed controls in this setting naturally solve an important difficulty, while being clearly interpretable in terms of geometry on the space of probability measures.